KEVOS® MAKEOS · Geometry
Spherical Sections: Areas and Volumes (Hollow Sphere, Sector, Segment, Zone, Ring)
What is the surface area, volume and mass of a sphere or of a part of one — a hollow sphere wall, a spherical sector, a segment (cap), a zone between two parallel planes, a hollow sector, a solid ring or part of a ring — and what radius of sphere do the given dimensions belong to?
Areas and volumes of a sphere and of the spherical sections drawn in the handbook: the whole sphere (or its radius from a volume), the wall G between two concentric spheres, the sector K of included angle φ, the segment S of base diameter E and height F, the zone T between parallel planes of diameters P and Q a height H apart (with the radius of its sphere), the hollow sector U between two spheres, the solid ring W (torus) and the part Z of a ring within an angle φ, with the mass from a density you enter.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Section | choice | Which spherical section to calculate, as lettered in the handbook figure; the fields each section uses are named in brackets. |
| Sphere | ||
| Sphere radius R | mm | Radius of the sphere for the whole sphere and the sector K. |
| Volume of the whole sphere (to find its radius) | L | Volume of a whole sphere when you want the radius it needs. |
| Wall, sector and ring | ||
| Outer radius R₁ (ring: radius to the centre of the ring section) | mm | Outer sphere radius for the wall G and hollow sector U; for the ring W and part ring Z, the radius from the ring axis to the centre of the ring section. |
| Inner radius R₂ (ring: radius of the ring section) | mm | Inner sphere radius for the wall G and hollow sector U; for the ring W and part ring Z, the radius of the circular ring section. |
| Included angle φ | ° | Included angle of the sector K or hollow sector U at the centre of the sphere, or the angle of the part ring Z about the ring axis. |
| Segment and zone | ||
| Base diameter E of the segment | mm | Diameter of the flat base of the segment (cap). |
| Height F of the segment | mm | Height of the segment from its base to the top of the sphere. |
| Height H of the zone | mm | Distance between the two parallel planes that bound the zone. |
| Diameter P at one end of the zone | mm | Diameter of the circle where one plane cuts the sphere. |
| Diameter Q at the other end of the zone | mm | Diameter of the circle where the other plane cuts the sphere. |
| Material | ||
| Density of the material (for the mass) | kg/m³ | Density of the material the section is made of, from your material data. |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Section
- What it is
- Spherical section to calculate, lettered as in the handbook figure of spherical areas and volumes.
- Units and format
- Choose one of 9 options from the list: Whole sphere (radius R); Whole sphere from its volume; Wall G between two concentric spheres (R₁, R₂); Sector K (R, angle φ); Segment (cap) S (E, F); Zone T between parallel planes (P, Q, H); Hollow sector U (R₁, R₂, angle φ); Solid ring W (R₁, R₂); Part Z of a ring (R₁, R₂, angle φ).
- Example
- The form opens with “Segment (cap) S (E, F)” selected as a worked example. Choose the option that matches your case.
- Effect on the result
- Chooses the formulas and which dimensions are used.
- Get it right
- A dished head or dome is a segment S; a band cut by two parallel planes is a zone T; a ring is W.
Sphere
What this means: Sphere radius R
- What it is
- Radius of the whole sphere, used for the whole sphere and the sector K.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 150 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Area grows with the square and volume with the cube of the radius.
- Get it right
- Half the measured diameter; for a sector, the radius of the spherical surface.
What this means: Volume of the whole sphere (to find its radius)
- What it is
- Volume of a whole sphere when you want the radius it needs.
- Units and format
- Enter a number in L: litres. The form accepts 0.000001 L or more. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 1 L as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the radius, area and mass for that section only.
- Get it right
- 1 L is 1,000,000 mm³; a 1 L sphere is about 124 mm in diameter.
Wall, sector and ring
What this means: Outer radius R₁ (ring: radius to the centre of…
- What it is
- Outer sphere radius for the wall G and hollow sector U; for a ring, the radius from the ring axis to the centre of its section.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 125 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Raises the area and volume of the wall, sector or ring.
- Get it right
- For a ring, measure the mean diameter of the ring and halve it.
What this means: Inner radius R₂ (ring: radius of the ring secti…
- What it is
- Inner sphere radius for the wall G and hollow sector U; for a ring, the radius of the circular ring section.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 100 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A larger inner radius leaves less material in the wall or sector; a larger section radius enlarges a ring.
- Get it right
- Must be less than R₁; for a ring it is half the thickness of the ring section.
What this means: Included angle φ
- What it is
- Included angle of the sector at the centre of the sphere, or the angle of a part ring about its axis.
- Units and format
- Enter a number in °: degrees of angle. The form accepts 0.000001° to 360°. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 90° as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Sets the depth and size of a sector or the share of a ring.
- Get it right
- A hemisphere is a sector of 180°; a quarter ring is 90°.
Segment and zone
What this means: Base diameter E of the segment
- What it is
- Diameter of the flat base of the spherical segment (cap).
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Wider bases give larger segments and flatter spheres.
- Get it right
- Measure across the open edge of a dished head or dome.
What this means: Height F of the segment
- What it is
- Height of the spherical segment from its flat base to the top of the dome.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- A higher segment holds more and comes from a smaller sphere.
- Get it right
- Measure square to the base at the centre.
What this means: Height H of the zone
- What it is
- Distance between the two parallel planes that bound the spherical zone.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 50 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Area of the zone is proportional to this height.
- Get it right
- Measure square to both cut faces.
What this means: Diameter P at one end of the zone
- What it is
- Diameter of the circle where one bounding plane cuts the sphere, at one end of the zone.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 280 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- With Q and H, fixes the radius of the sphere.
- Get it right
- Measure across one cut face of the band.
What this means: Diameter Q at the other end of the zone
- What it is
- Diameter of the circle where the other bounding plane cuts the sphere, at the other end of the zone.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 1000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 200 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- With P and H, fixes the radius of the sphere.
- Get it right
- Measure across the other cut face of the band.
Material
What this means: Density of the material (for the mass)
- What it is
- Density of the material the spherical section is made of, for its mass.
- Units and format
- Enter a number in kg/m³: kilograms per cubic metre. The form accepts 0.000001 kg/m³ to 30000 kg/m³. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 7850 kg/m³ as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Mass is proportional to density; areas and volumes do not change.
- Get it right
- Steel is about 7850 kg/m³ and aluminium about 2700 kg/m³; use your material data.
Method and formulas
- Sphere: area 4πR², volume (4/3)πR³; radius from volume R = ∛(3V ÷ 4π).
- Wall G: area 4πR₁² + 4πR₂², volume (4/3)π(R₁³ − R₂³).
- Sector K: area of the spherical surface 2πR²(1 − cos ½φ), depth F = R(1 − cos ½φ), volume 2πR²F ÷ 3.
- Segment S: area π(F² + E²/4), volume πF(E²/8 + F²/6); sphere radius (E²/4 + F²) ÷ 2F.
- Zone T: sphere radius √(((P² − Q² − 4H²) ÷ 8H)² + P²/4), area 2πRH, volume H(π/6)(H² + 3Q²/4 + 3P²/4).
- Hollow sector U: area 2π(R₁² + R₂²)(1 − cos ½φ), volume (2π/3)(R₁³ − R₂³)(1 − cos ½φ), the sector K of R₁ less that of R₂.
- Ring W: area 4π²R₁R₂, volume 2π²R₁R₂²; part ring Z: those values × φ ÷ 360.
- Example: a segment 200 mm across its base and 50 mm high has a volume of 850,850 mm³ and is cut from a sphere of radius 125 mm.
Limitations
- Areas are of the curved spherical or ring surfaces the handbook lists; flat ends and the conical faces of sectors are not included.
- The hollow sector volume uses (2π/3), consistent with the sector K formula; the handbook table prints 2π, which overstates the volume three times.
- Mass uses the density you enter and the volume of a solid section.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the volume and mass of a dished head, cap or domed part
- Find the radius of the sphere a zone or cap was cut from
- Work out the material in a hollow ball, ring or part ring
Who it is for
- Designers and drafters
- Estimators
- Fabricators
- Students
What you get
- Surface area of the section
- Volume of the section
- Mass at the density entered
- Radius of the sphere (from a volume, segment or zone)
- Depth F of a sector from its angle
Interpreting the result
- A segment higher than the radius of its sphere is more than a hemisphere; the same formulas hold.
- For a zone, the two planes can be on the same side of the centre or on opposite sides; the radius formula covers both.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 52–54. Topic reference; no source tables or recommended settings are reproduced.
Learn the Concepts Behind This Tool
These KEVOS® Library readings explain the method this tool uses, with worked examples.
Method explained here · KEVOS® original book
Sizing the blank for a dome
In Hand Forming Sheet Metal by the Numbers
What it covers: Size a dome's blank by sinking, raising or both, predict the mean thickness and the rim's shrink for each, make a checking template, and catch a blank cut with no trim allowance.
Method explained here · KEVOS® original book
Cones, pyramids and frustums
In Tanks and Hoppers by the Numbers
What it covers: Calculate cone and pyramid bottoms and the frustums that hoppers are made of, find wall and hip slopes, and catch an average-of-the-ends estimate that overstated a hopper by 21%.
KEVOS® tools are screening and decision-support utilities. Verify important financial, engineering, safety, legal or operational decisions using appropriate professional standards and your organisation’s approved data.